Computer aided design system and computer aided design program using a geometric surface model

ABSTRACT

A computer aided design system and a computer aided design program which can greatly increase the utility of a computer aided design model, and can improve the efficiency of design and production processes, by adopting a curved surface theory which ensures the continuity of a free-form line or surface. A computer executes: a point sequence information extraction process for extracting a plurality of point sequences on a curved surface; a dividing process for generating a curved surface from the point sequences and dividing the curved surface into a predetermined number of mesh points; a first fundamental form computing process for computing coefficients of the first fundamental form; a second fundamental form computing process for computing coefficients of the second fundamental form; and a storage process for storing the point sequence information, the coefficients of the first fundamental form and the coefficients of the second fundamental form.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to a computer aided design system and acomputer aided design program which transform the shape of a member intoan objective curved surface shape.

2. Background Art

Today, there is a desire to shorten processes from planning to designand production to respond to consumer demand. In order to improve theefficiency of design and production processes, the use of CG (ComputerGraphics) and CAD (Computer Aided Design) systems is popular. In orderto depict shapes having complex curved lines or curved surface shapes,such as for motor vehicles, domestic electric appliances, or the like,on a computer, the following processing methods have conventionally beenused.

The first method is solid modeling, where simple shapes calledprimitives are held in a computer, and operations to combine the shapeswith each other are repeated in order to express complex shapes. Aprimitive is for example a column, a cube, a hexahedron, a torus, aball, or the like, and in solid modeling shapes are represented by setoperations on these primitives. Therefore, in order to produce a complexshape many steps are required and precise calculations are required.

The second method is surface modeling, which utilizes an algorithm suchas a bezier, b-spline, rational bezier, NURBS (Non-Uniform Rationalb-spline), or the like in order to perform operations such as cutting orconnecting lines or surfaces, and by repeating these operations complexfree curved lines or curved surfaces are represented.

However, even with a model in which there are no problems in therepresentation with the solid model or surface model described above, insome cases problems may occur when the model is used by a downstreamapplication such as CAM, CAE, or the like. This is caused by differencesbetween the support element to be supported by the produced computergraphics, and the support element to be supported by the other computergraphics, computer aided design, and downstream applications, anddifferences in shape definition, or the like. The model is corrected viaan application such as a translator which modifies these differences(Japanese Unexamined Patent Publication Nos. 2001-250130, Hei 11-65628,Hei 10-69506, Hei 4-134571, Hei 4-117572, and Hei 1-65628).

BRIEF SUMMARY OF THE INVENTION

However, the above described correcting operations greatly prolong thedesign and production processes. The reasons for requiring thecorrections vary depending on each case, but the point which becomes aproblem, particularly in the production stage is that therepresentations of all curved lines and curved surfaces are approximatedby Euclidean geometry in a conventional computer graphics or computeraided design system. For example, in the case where a saddle-typesurface of the tabulated surfaces shown in FIG. 6 is generated by asweep operation, there exists a long line in the lower slope part of thesaddle and a short line in the central part of the saddle. Therefore,this sweep operation is a transformation accompanied with graphicalexpansion and contraction in order to maintain the continuity of thecurved surface generated. However, in a conventional computer graphicsor computer aided design system, this expansion and contraction is notconsidered, and the internal representation is approximately representedas a cylinder type. Therefore, if the computer graphics model orcomputer aided design model which are approximately represented by suchEuclidean geometry, are passed to a CAE, the error occurring in themodel becomes a problem in production.

The present invention has been devised to solve such problems, with anobject of providing a computer aided design system and a computer aideddesign program which can efficiently utilize a computer graphics modelor a computer aided design model, and can improve the efficiency ofdesign and production processes.

A computer aided design system of the present invention comprises: apoint sequence information extraction device which extracts a pluralityof point sequences on a curved surface; a dividing device whichgenerates a curved surface from the point sequences using anothercomputer aided design system, and divides the curved surface into apredetermined number of meshes; a first fundamental form computingdevice for computing coefficients of the first fundamental form definedby a tangent vector which forms a tangent plane of the mesh; a secondfundamental form computing device for computing coefficients of thesecond fundamental form defined by the tangent vector and a normalvector of the mesh; and a memory device which stores the point sequenceinformation, the coefficients of the first fundamental form and thecoefficients of the second fundamental form.

Furthermore, the computer aided design system of the present inventionfurther comprises: a principal curvature computing device which computesa principal curvature of the mesh based on the coefficients of the firstfundamental form and the coefficients of the second fundamental form; aline of curvature computing device which computes a line of curvatureshowing a principal direction of the mesh based on the principalcurvature; a feature point/feature line analyzing device which extractsa point or a line which become a reference point or a reference line oftransformation defined by changing patterns of one or more featurequantities among five feature quantities showing features of the curvedsurface comprising a Gaussian curvature and a mean curvature computedbased on the principal curvature, the principal direction, the line ofcurvature, and the coefficients of the first fundamental form andcoefficients of the second fundamental form; and a girth lengthcomputing device which computes a girth length based on a curvaturecomputed from the coefficients of the first fundamental form and thecoefficients of the second fundamental form.

Moreover, the computer aided design system of the present inventionfurther comprises: a reproducing device which transforms the line ofcurvature for the girth length in the line of curvature direction, withthe feature point or feature line as a transformation reference, andreproduces the mesh or the curved surface.

Furthermore, the computer aided design system of the present inventionfurther comprises: a converting device which extracts a plurality ofpoint sequences on a curved surface from the reproduced mesh or curvedsurface, and converts the point sequences according to a graphicalrepresentation algorithm in another computer aided design system.

A computer aided design program of the present invention executes on acomputer: a point sequence information extraction process for extractinga plurality of point sequences on a curved surface; a dividing processfor generating a curved surface from the point sequences using anothercomputer aided design system, and dividing the curved surface into apredetermined number of meshes; a first fundamental form computingprocess for computing coefficients of the first fundamental form definedby a tangent vector which forms a tangent plane of the mesh; a secondfundamental form computing process for computing coefficients of thesecond fundamental form defined by the tangent vector and a normalvector of the mesh; and a storage process for storing the point sequenceinformation, the coefficients of the first fundamental form and thecoefficients of the second fundamental form.

Moreover, the computer aided design program of the present invention isa computer aided design program for further executing on a computer: aprincipal curvature computing process for computing a principalcurvature of the mesh based on the coefficients of the first fundamentalform and the coefficients of the second fundamental form; a line ofcurvature computing process for computing a line of curvature showing aprincipal direction of the mesh based on the principal curvature; afeature point/feature line analyzing process for extracting a point or aline which become a reference point or a reference line oftransformation defined by changing patterns of one or more featurequantities among five feature quantities showing features of the curvedsurface comprising a Gaussian curvature and a mean curvature computedbased on the principal curvature, the principal direction, the line ofcurvature, and the coefficients of the first fundamental form and thecoefficients of the second fundamental form; and a girth lengthcomputing process for computing a girth length based on a curvaturecomputed from the coefficients of the first fundamental form andcoefficients of the second fundamental form.

Furthermore, the computer aided design program of the present inventionis a computer aided design program for further executing on a computer:a reproducing process for transforming the line of curvature for thegirth length in the line of curvature direction, with the feature pointor feature line as a transformation reference, and reproducing the meshor the curved surface.

Moreover, the computer aided design program of the present invention isa computer aided design program for further executing on a computer: aconverting process for extracting a plurality of point sequences on acurved surface from the reproduced mesh or curved surface, andconverting the point sequences according to a graphical representationalgorithm in another computer aided design system.

A computer graphics system of the present invention comprises: a pointsequence information extraction device which extracts a plurality ofpoint sequences on a curved surface; a dividing device which generates acurved surface from the point sequences using another computer graphicssystem, and divides the curved surface into a predetermined number ofmeshes; a first fundamental form computing device for computingcoefficients of the first fundamental form defined by a tangent vectorwhich forms a tangent plane of the mesh; a second fundamental formcomputing device for computing coefficients of the second fundamentalform defined by the tangent vector and a normal vector of the mesh; anda memory device which stores the point sequence information, thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form.

Moreover, the computer graphics program of the present invention is acomputer graphics program for executing on a computer: a point sequenceinformation extraction process for extracting a plurality of pointsequences on a curved surface; a dividing process for generating acurved surface from the point sequences using another computer graphicssystem, and dividing the curved surface into a predetermined number ofmeshes; a first fundamental form computing process for computingcoefficients of the first fundamental form defined by a tangent vectorwhich forms a tangent plane of the mesh; a second fundamental formcomputing process for computing coefficients of the second fundamentalform defined by the tangent vector and a normal vector of the mesh; anda storage process for storing the point sequence information, thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form.

The present invention demonstrates the following effects.

Since it comprises: the point sequence information extraction devicewhich extracts a plurality of point sequences on a curved surface; thedividing device which generates a curved surface from the pointsequences using another computer graphics system or computer aideddesign system, and divides the curved surface into a predeterminednumber of meshes; the first fundamental form computing device forcomputing coefficients of the first fundamental form defined by atangent vector which forms a tangent plane of the mesh; the secondfundamental form computing device for computing coefficients of thesecond fundamental form defined by the tangent vector and a normalvector of the mesh; and the memory device which stores the pointsequence information, the coefficients of the first fundamental form andthe coefficients of the second fundamental form, then by adopting acurved surface theory which ensures the continuity of a free-formline/free-form surface, a computer graphics model or a computer aideddesign model can be widely utilized, and the efficiency of design andproduction processes can be improved.

Moreover, since it further comprises: the principal curvature computingdevice which computes a principal curvature of the mesh based on thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form; the line of curvature computing device whichcomputes a line of curvature showing a principal direction of the meshbased on the principal curvature; the feature point/feature lineanalyzing device which extracts a point or a line which become areference point or a reference line of transformation defined bychanging patterns of one or more feature quantities among five featurequantities showing features of the curved surface comprising a Gaussiancurvature and a mean curvature computed based on the principalcurvature, the principal direction, the line of curvature, and thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form; and the girth length computing device whichcomputes a girth length based on a curvature computed from thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form, then a computer graphics or computer aideddesign model analyzed by the curved surface theory can be reproduced andconverted into another computer graphics or computer aided design model.

Furthermore, since it further comprises: the reproducing device whichtransforms the line of curvature for the girth length in the line ofcurvature direction, with the feature point or the feature line as thetransformation reference, and reproduces the mesh or the curved surface,then a computer graphics or computer aided design model analyzed by thecurved surface theory can be reproduced.

Moreover, since it further comprises: the converting device whichextracts a plurality of point sequences on a curved surface from thereproduced mesh or curved surface, and converts the point sequencesaccording to a graphic expression algorithm in another computer graphicsor computer aided design system, then a computer graphics or computeraided design model analyzed by the curved surface theory can beconverted into another computer graphics or computer aided design model.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram showing a configuration of a computer aideddesign system of the present embodiment.

FIG. 2 is an explanatory diagram showing a situation for dividing acurved surface into an m×n mesh and defining fundamental vectors Su andSv.

FIG. 3 is an explanatory diagram showing planes in which a unit tangentvector t and an unit normal vector n extend.

FIG. 4 is a flowchart showing a processing flow from free-form surfaceanalysis to data transfer, by an analysis program 1.

FIG. 5 is an explanatory diagram showing an aspect of curvature change.

FIG. 6 is an explanatory diagram showing classifications of meancurvature and Gaussian curvature.

FIG. 7 is an explanatory diagram showing isoclinic orthogonal lines.

FIG. 8 is an explanatory diagram showing principal curvature extremumlines.

FIG. 9 is an explanatory diagram showing isoclinic extremum lines andaspects of Gaussian curvature distribution.

FIG. 10 is an explanatory diagram showing lines of curvature.

DETAILED DESCRIPTION OF THE INVENTION

Hereunder is a description of an embodiment of a computer aided designsystem of the present invention, with reference to the drawings. FIG. 1is a block diagram showing a configuration of the computer aided designsystem of the present embodiment. The computer aided design system ofthe present embodiment comprises a central processor such as centralprocessing unit (CPU) or the like (not shown), a storage memory such asa ROM, RAM, or the like (not shown), a database 10, a graphic displayprocessing section 11, a display section 12, an output section 13, and acommunication section (not shown).

The CPU reads out an analysis program 1, a converting program 2, and areproducing program 3 stored in a ROM and executes a series of processesrelated to free-form surface analysis, conversion, and reproduction. TheRAM is a semiconductor memory in which the CPU primarily stores data.

The analysis program 1 is a computer program which executes a processfor reading in actual measurement value data 20 of a three-dimensionalshaped body by CAT (computer aided testing) or the like, or othercomputer aided design format data 21 (graphics data represented forexample by a surface model such as a solid model, a bezier, a b-spline,a rational bezier, or a NURBS), creating a point sequence informationtable 30, a table 31 of coefficients of the first fundamental form, anda table 32 of coefficients of the second fundamental form, and thenstoring this in the database 10.

The point sequence information table 30 comprises point sequenceinformation (u, v) for on a curved surface expressed in a parameter formof;s(u,v)={x(u,v),y(u,v),z(u,v)} 0<u, v<1  (equation 1)

as shown in FIG. 2. For example, assuming that u=0, 1/m, 2/m, to m−1/m(m is a natural number) and v=0, 1/n, 2/n, to n−1/n (n is a naturalnumber), the curved surface shown in FIG. 2 is divided into an m×n mesh.In this case, the point sequence information (u, v) becomes an mn datasequence from the mesh ID1 to IDmn.

The table 31 of coefficients of the first fundamental form comprisescoefficients of the first fundamental form E, F, and G derived from thefollowing equations. In the case where u and v described above have afunctional relation, then (u, v) denotes a curved line on the curvedsurface, the partial derivative ∂s/∂u=Su denotes a tangent vector of acurved line of u=constant, and the partial derivative ∂s/∂v=Sv denotes atangent vector of a curved line of v=constant. At this time, thefundamental vectors Su and Sv form a tangent plane of the curvedsurface. Moreover, a vector ds linking two points on the curved surface,from s (u, v) to s (u+du, v+dv), is represented by:ds=s _(u) du+s _(v) dv  (equation 2)

Here, the square of the absolute value of ds is represented by:(ds)² =ds·ds=s _(u) ²(du)²+2s _(u) ·s _(v) dudv+s _(v) ²(dv)²  (equation3)

The coefficients of the first fundamental form described above aredefined from the fundamental vector of the curved surface by thefollowing equation:E=s _(u) ² , F=s _(u) ·s _(v) , G=s _(v) ²  (equation 4)

The coefficients of the first fundamental form E, F, and G describedabove are uniquely determined for the respective mesh points in thisway. The table 31 of coefficients of the first fundamental form storesthe values for the respective mesh points ID1 to IDmn.

Moreover, combining the abovementioned equation 3 and equation 4 gives:ds ² =E(du)²+2Fdudv+G(dv)²  (equation 5)

The table 32 of coefficients of the second fundamental form comprisescoefficients of the second fundamental form L, M, and N derived from thefollowing equations. Assuming that ω is an angle between the fundamentalvectors Su and Sv, then their inner product F, and the absolute value Hof the vector product of the fundamental vectors, are represented asfollows using the coefficients of the first fundamental form:F=|s _(u) |·|s _(v)| cos ω  (equation 6)

$\begin{matrix}\begin{matrix}{H = {{s_{u} \times s_{v}}}} \\{= {{{s_{u}} \cdot {s_{v}}}\sin\;\omega}} \\{= \sqrt{{EG}\left( {1 - {\cos^{2}\omega}} \right)}} \\{= \sqrt{{EG} - F^{2}}}\end{matrix} & \left( {{equation}\mspace{14mu} 7} \right)\end{matrix}$

Then using this calculated value H, the unit normal vector n on thecurved surface is represented by:

$\begin{matrix}{n = \frac{\left( {s_{u} \times s_{v}} \right)}{H}} & \left( {{equation}\mspace{14mu} 8} \right)\end{matrix}$

Moreover, as shown in FIG. 3, the pencil of lines of the tangent vectorsat a point P on the curved surface exists in this tangent plane, and aunit tangent vector t of these is represented by the following equation:

$\begin{matrix}{t = {\frac{\mathbb{d}s}{\mathbb{d}s} = {{s_{u}\left( \frac{\mathbb{d}u}{\mathbb{d}s} \right)} + {s_{v}\left( \frac{\mathbb{d}v}{\mathbb{d}s} \right)}}}} & \left( {{equation}\mspace{14mu} 9} \right)\end{matrix}$

The plane determined by t and n as shown in FIG. 3 is called a normalplane.

The curvature κ at the point P on this normal section plane is called anormal curvature. Differentiating t for the arc length s on the normalsection plane gives:

$\begin{matrix}{\frac{\mathbb{d}t}{\mathbb{d}s} = {{s_{u}\frac{\mathbb{d}^{2}u}{\mathbb{d}s^{2}}} + {s_{v}\frac{\mathbb{d}^{2}v}{\mathbb{d}s^{2}}} + {s_{uu}\left( \frac{\mathbb{d}u}{\mathbb{d}s} \right)}^{2} + {2{s_{uv}\left( \frac{\mathbb{d}u}{\mathbb{d}s} \right)}\left( \frac{\mathbb{d}v}{\mathbb{d}s} \right)} + {s_{vv}\left( \frac{\mathbb{d}v}{\mathbb{d}s} \right)}^{2}}} & \left( {{equation}\mspace{14mu} 10} \right)\end{matrix}$

Multiplying both equations by the normal vector, and introducing thefollowing coefficients of the second fundamental form:L=n·s _(uu) , M=n·s _(uv) , N=n·s _(vv)  (equation 11)

gives:

$\begin{matrix}{{\left( {n \cdot n} \right)\kappa} = {{L\left( \frac{\mathbb{d}u}{\mathbb{d}s} \right)}^{2} + {2{M\left( \frac{\mathbb{d}u}{\mathbb{d}s} \right)}\left( \frac{\mathbb{d}v}{\mathbb{d}s} \right)} + {N\left( \frac{\mathbb{d}v}{\mathbb{d}s} \right)}^{2}}} & \left( {{equation}\mspace{14mu} 12} \right)\end{matrix}$

The coefficients of the second fundamental form L, M, and N describedabove are uniquely determined for the respective meshes in this way. Thetable 32 of coefficients of the second fundamental form stores thevalues for the respective points ID1 to IDmn.

If equation 5 is substituted in equation 12, the following equation isobtained.

$\begin{matrix}{\kappa = \frac{{L\left( {\mathbb{d}u} \right)}^{2} + {2M{\mathbb{d}u}{\mathbb{d}v}} + {N\left( {\mathbb{d}v} \right)}^{2}}{{E\left( {\mathbb{d}u} \right)}^{2} + {2F{\mathbb{d}u}{\mathbb{d}v}} + {F\left( {\mathbb{d}v} \right)}^{2}}} & \left( {{equation}\mspace{14mu} 13} \right)\end{matrix}$

From the above, the normal curvature is computed from the coefficientsof the first fundamental form and the coefficients of the secondfundamental form.

The converting program 2 is a computer program which executes a processfor reading out the necessary information for a free-form surface fromthe point sequence information table 30, the table 31 of coefficients ofthe first fundamental form, and the table 32 of coefficients of thesecond fundamental form, and then creating free-form surface data, andtransforming this into a form which another computer aided designapplication can interpret.

The reproducing program 3, similarly to the converting program 2, is acomputer program which executes a process for reading out the necessaryinformation for a free-form surface from the point sequence informationtable 30, the table 31 of coefficients of the first fundamental form,and the table 32 of coefficients of the second fundamental form, thencreating free-form surface data, and outputting to the graphic displayprocessing section 11.

The database 10 stores the above described point sequence informationtable 30, the table 31 of coefficients of the first fundamental form,and the table 32 of coefficients of the second fundamental form, andwrites in the output result of the analysis program 1 in associationwith a mesh ID described later.

The graphic display processing section 11 performs graphic displayprocessing on the output results from the reproducing program, and othercomputer aided design applications.

The display section 12 displays the output results of the graphicdisplay processing section 11.

The output section 13 outputs the output results of the graphic displayprocessing section 11 to the communication section, other recordingmedia, or the like. The communication section transfers data such as thepoint sequence information, the coefficients of the first fundamentalform, and the coefficients of the second fundamental form stored in thedatabase 1 to other severs or clients via a network such as a LAN, theInternet, or the like.

Next is a description of a series of processing flows related to thefree-form surface analysis, conversion, and reproduction by the computeraided design system of the present embodiment, with reference to thedrawings. FIG. 4 is a flowchart showing a processing flow from free-formsurface analysis to data transfer, by the analysis program 1.

By the user's operation, the CPU receives an analyze command for theactual measurement value data 20 or other computer aided design formatdata 21, reads out the analysis program 1 from ROM, and executes thefree-form surface analyzing process. Firstly, the CPU performs a processfor extracting a plurality of point sequences on a curved surface suchas a two-dimensional NURBS surface, bicubic surface, or the like, heldby the actual measurement value data 20 or the other computer aideddesign format data 21. Then, a curved surface is generated from thispoint sequence using the other computer aided design system (step S1 inFIG. 4), and the curved surface is divided into a predetermined mn meshpoints as shown in FIG. 2, after which the respective mesh parts arestandardized by fundamental vectors Su and Sv. The point sequenceinformation (u, v) generated during the standardization is written inthe point sequence information table 30 held by the database 10, inassociation with the mesh ID.

Next, the CPU executes differential geometric analysis processing. Thatis, it performs processing for computing the coefficients of the firstfundamental form E, F, and G defined by the fundamental vectors Su andSv which form the tangent plane of the mesh. The computed coefficientsof the first fundamental form E, F and G, similarly to the pointsequence information, are written in the table 31 of coefficients of thefirst fundamental form held by the database 10, in association with themesh ID. Moreover, the CPU performs a process for computing thecoefficients of the second fundamental form L, M, and N defined by thefundamental vectors Su and Sv and an unit normal vector n of the mesh.The computed coefficients of the second fundamental form L, M, and N,similarly to the coefficients of the first fundamental form E, F and G,are written in the table 32 of coefficients of the second fundamentalform held by the database 10, in association with the mesh ID.

Moreover, the CPU performs processing for computing an integrablecondition which is a condition where the differential equationrepresenting the above described mesh is continuous at the respectiveboundaries of the mesh, in other words, a condition where thisdifferential equation has a unique solution.

Now, the curved surface coordinates (u, v) described above aresubstituted by (u1, u2) and the point is made p (u1, u2). If a curvedline formed by fixing u2 and moving u1 is called a u1 curved line, and acurved line formed by fixing u1 and moving u2 is called a u2 curvedline, then assuming that p (u1, u2) on the curved surface is the initialpoint, then the tangent vector along the u1 curved line and the u2curved line can be calculated as follows:

$\begin{matrix}{{e_{1} = \frac{\partial p}{\partial u^{1}}},{e_{2} = \frac{\partial p}{\partial u^{2}}}} & \left( {{equation}\mspace{14mu} 14} \right)\end{matrix}$

Then, the unit normal vector n can be calculated from e1 and e2 asfollows:

$\begin{matrix}{n = \frac{e_{1} \times e_{2}}{{e_{1} \times e_{2}}}} & \left( {{equation}\mspace{14mu} 15} \right)\end{matrix}$

In this way, three vectors {e1, e2, n} are defined for the respectivepoints on the curved surface.

For the respective points, first fundamental quantities E, F and G aredefined as follows:E=∥e ₁∥² , F=(e ₁ ,e ₂), G=∥e ₂∥²  (equation 16)

Then, a first fundamental tensor (g_(ij), i, j=1, 2) is defined asfollows:g₁₁=E, g₁₂=g₂₁=F, g₂₂=G  (equation 17)

Moreover, four numerical sets g^(ij), i, j=1, 2 are defined as follows.

$\begin{matrix}{{g^{11} = \frac{G}{{EG} - F^{2}}},{g^{12} = {g^{21} = {- \frac{F}{{EG} - F^{2}}}}},{g^{22} = \frac{E}{{EG} - F^{2}}}} & \left( {{equation}\mspace{14mu} 18} \right)\end{matrix}$

Furthermore, for the respective points, second fundamental quantities L,M and N are defined as follows:

$\begin{matrix}{{L = \left( {\frac{\partial^{2}p}{\partial\left( u^{1} \right)^{2}},n} \right)},{M = \left( {\frac{\partial^{2}p}{{\partial u^{1}}{\partial u^{2}}},n} \right)},{N = \left( {\frac{\partial^{2}p}{\partial\left( u^{2} \right)^{2}},n} \right)}} & \left( {{equation}\mspace{14mu} 19} \right)\end{matrix}$

Then, a second fundamental tensor (h_(ij), i, j=1, 2) is defined asfollows:h₁₁=L, h₁₂=h₂₁=M, h₂₂=N  (equation 20)

Now, if the dynamic frame {e1, e2, n} is differentiated by the curvedsurface coordinates (u1, u2), structural equations of a curved surfaceshown by the following two equations (Gaussian equation of equation 21and Weingarten's equation of equation 22) are obtained:

$\begin{matrix}{\frac{\partial e^{i}}{\partial u^{j}} = {{\begin{Bmatrix}k & \; \\i & j\end{Bmatrix}e_{k}} + {h_{ij}n}}} & \left( {{equation}\mspace{14mu} 21} \right) \\{\frac{\partial n}{\partial u^{i}} = {{- g^{jk}}h_{ij}e_{k}}} & \left( {{equation}\mspace{14mu} 22} \right) \\{\begin{Bmatrix}k & \; \\i & j\end{Bmatrix} = {\frac{1}{2}{g^{kl}\left( {\frac{\partial g_{lj}}{\partial u^{i}} + \frac{\partial g_{li}}{\partial u^{j}} + \frac{\partial g_{ij}}{\partial u^{l}}} \right)}}} & \left( {{equation}\mspace{14mu} 23} \right)\end{matrix}$where equation 23 exhibits the Christoffel symbol.

The integrable condition of these structural equations 21 and 22 isshown by the following two equations (the Gaussian equation of equation24 and the Mainardi-Codazzi's equation of equation 25):

$\begin{matrix}{R_{jkl}^{i} = {g^{im}\left( {{h_{jk}h_{l\; m}} - {h_{jl}h_{k\; m}}} \right)}} & \left( {{equation}\mspace{14mu} 24} \right) \\{{\frac{\partial h_{ij}}{\partial u^{k}} - \frac{\partial h_{ik}}{\partial u^{j}} + {\begin{Bmatrix}l & \; \\i & j\end{Bmatrix}h_{lk}} - {\begin{Bmatrix}l & \; \\i & k\end{Bmatrix}h_{lj}}} = 0} & \left( {{equation}\mspace{14mu} 25} \right) \\{R_{jkl}^{i} = {{\frac{\partial\;}{\partial u^{l}}\begin{Bmatrix}l & \; \\j & k\end{Bmatrix}} - {\frac{\partial\;}{\partial u^{k}}\begin{Bmatrix}i & \; \\j & l\end{Bmatrix}} + {\begin{Bmatrix}m & \; \\j & k\end{Bmatrix}\begin{Bmatrix}i & \; \\m & l\end{Bmatrix}} - {\begin{Bmatrix}m & \; \\j & l\end{Bmatrix}\begin{Bmatrix}i & \; \\m & k\end{Bmatrix}}}} & \left( {{equation}\mspace{14mu} 26} \right)\end{matrix}$where equation 26 exhibits the Riemann-Christoffel curvature tensor.

In the case where the first fundamental tensor (g_(ij), i, j=1, 2) andthe second fundamental tensor (h_(ij), i, j=1, 2) are applied asfunctions of the curved surface coordinate (u1, u2) and they satisfy theGaussian equation and the Mainardi-Codazzi's equation described above,the shape of the curved surface having such g_(ij) and h_(ij) isuniquely determined (refer to Bonnet's fundamental theory). Therefore,the mesh is C2 continuous.

The CPU performs these arithmetic processes and calculates theintegrable condition described above (step S2).

Next, the CPU executes a line of curvature analyzing process, a featureline analyzing process, and a curvature/girth length converting process(step S3). Firstly, the principal curvatures κ₁ and κ₂ in the mesh arecalculated based on the coefficients of the first fundamental form E, Fand G, and the coefficients of the second fundamental form L, M and N bythe line of curvature analyzing process (step S4). That is to say,firstly the extremum of the curvature κ₁ described above is calculated.The shape of the normal section plane, which is the line of intersectionof the normal plane and the curved surface, changes together with thetangential direction, and accompanied by this, the normal curvature alsochanges. This shape returns to the initial condition when the normalplane is half rotated. Now, assuming that

$\begin{matrix}{\gamma = \frac{\mathbb{d}v}{\mathbb{d}u}} & \left( {{equation}\mspace{14mu} 27} \right)\end{matrix}$and rewriting i as the function κ(γ) of γ, gives:{L−κ(γ)·E}+2{M−κ(γ)·F}γ+{N−κ(γ)·G}γ ²=0  (equation 28)

From this quadratic equation of γ, for dκ(γ)/dγ=0, κ(γ) becomes theextremum. Then, if equation 15 is differentiated in this condition forthe extremum, and κ and γ are rewritten as ({tilde over (κ)}) and({tilde over (γ)}),(M−κ{tilde over ( )}F)+(N−κ{tilde over ( )}G)γ{tilde over ()}=0  (equation 29)is obtained. Then, if it is substituted in equation 16,(L−κ{tilde over ( )}E)+(M−κ{tilde over ( )})γ{tilde over ()}=0  (equation 30)is obtained. The following relations are obtained from these equations:

$\begin{matrix}{\overset{\sim}{\gamma} = {\frac{- \left( {M - {\overset{\sim}{\kappa}\; F}} \right)}{\left( {N - {\overset{\sim}{\kappa}\; G}} \right)} = \frac{- \left( {L - {\overset{\sim}{\kappa}\; E}} \right)}{\left( {M - {\overset{\sim}{\kappa}\; F}} \right)}}} & \left( {{equation}\mspace{14mu} 31} \right) \\{\overset{\sim}{\kappa} = {\frac{\left( {M + {\overset{\sim}{\gamma}\; N}} \right)}{\left( {F + {\overset{\sim}{\gamma}\; G}} \right)} = \frac{\left( {L + {\overset{\sim}{\gamma}\; M}} \right)}{\left( {E + {\overset{\sim}{\gamma}\; F}} \right)}}} & \left( {{equation}\mspace{14mu} 32} \right)\end{matrix}$

If equation 18 is modified,(EG−F ²)κ{tilde over ( )}²−(EN+LG−2MF)κ{tilde over ( )}+LN−M²=0  (equation 33)is obtained. The coefficient of {tilde over (κ)}² is positive fromequation 7. Assuming that the roots are κ₁ and κ₂, the value becomes theprincipal curvature as shown in FIG. 5.

Next, the Gaussian curvature or the mean curvature are calculated basedon the principal curvature (step S5). That is, from the relation of theroots and the coefficients of the quadratic equation,

$\begin{matrix}{K_{m} = {{\frac{1}{2}\left( {\kappa_{1} + \kappa_{2}} \right)} = {\frac{1}{2}\frac{\left( {{EN} + {LG} - {2{MF}}} \right)}{\left( {{EG} - F^{2}} \right)}}}} & \left( {{equation}\mspace{14mu} 34} \right) \\{K_{g} = {{\kappa_{1}\kappa_{2}} = \frac{\left( {{L\; N} - M^{2}} \right)}{\left( {{EG} - F^{2}} \right)}}} & \left( {{equation}\mspace{14mu} 35} \right)\end{matrix}$are calculated. Here, K_(m) is the mean curvature and K_(g) is theGaussian curvature. When K_(g)=0, this is the case where the curvedsurface becomes the developable surface as shown in FIG. 6, and the lineof curvature on the curved surface becomes a straight line. In thepresent embodiment, the point where this Gaussian curvature becomes zerois assumed to be the reference point of transformation described later.

As an appropriate point for the reference point of transformation otherthan this point, for example, lines of curvature, borderlines(ridgelines), isoclinic orthogonal lines shown in FIG. 7, principalcurvature extremum lines shown in FIG. 8, isoclinic extremum lines shownin FIG. 9, or umbilicus points may be selected. These are points orlines which become a reference point or a reference line oftransformation defined by changing patterns of one or more featurequantities among the principal curvature, the principal direction, theGaussian curvature, the mean curvature and the line of curvature, whichare feature quantities showing the feature of the curved surface. It ispossible to calculate these based on the coefficients of the firstfundamental form and the coefficients of the second fundamental form.

Moreover, the line of curvature showing the principal direction of themesh is calculated based on the principal curvature. That is,eliminating {tilde over (κ)} from equation 19 gives:(MG−NF)γ{tilde over ( )}²+(GL−NE)γ{tilde over ( )}+FL−ME=0  (equation36)or(MG−NF)dv ²+(GL−NE)dudv+(FL−ME)du ²=0  (equation 37)

Both these equations are equations of the line of curvature and thequadratic equation so that γ₁ and γ₂ have the following relations:

$\begin{matrix}{{{\gamma_{1} + \gamma_{2}} = \frac{- \left( {{GL} - {NE}} \right)}{\left( {{MG} - {NF}} \right)}},{{\gamma_{1}\gamma_{2}} = \frac{\left( {{FL} - {ME}} \right)}{\left( {{MG} - {NF}} \right)}}} & \left( {{equation}\mspace{14mu} 38} \right)\end{matrix}$

At a point on the curved surface, the curvature becomes an extremum inthe direction determined by γ₁ and γ₂. The tangent vector on the curvedsurface is (Sudu+Svdv) and the inner product of the two tangent vectorscorresponding to γ₁ and γ₂ becomes:(ds)₁·(ds)₂={(s _(u) +s _(v)γ₁)·(s _(u) +s _(v)γ₂)}(du)₁(dv)₂  (equation39)If inside the bracket { } is converted then:{E(MG−NF)−F(GL−NE)+G(FL−ME)}/(MG−NF)  (equation 40)becomes zero. That is, it is found that the two tangential directions ofthe normal section plane of the principal curvature become orthogonal.This direction is called the principal direction. In the case where thisdirection and the tangent line on the curved surface are matched, thisbecomes the lines of curvature shown in FIG. 10.

From the above, the calculation process of the line of curvature showingthe principal direction of the mesh is performed.

Next, the curvature/girth length converting process is performed (stepS6). That is, the CPU calculates the girth length based on the curvaturewhich is calculated based on the coefficients of the first fundamentalform E, F, and G and the coefficients of the second fundamental form L,M, and N. Along the line of curvature calculated by the line ofcurvature calculating process described above, the radius of curvatureis calculated from the curvature (1/r), and the girth length of the lineof curvature is expanded and contracted in each calculation interval.

From the above, the analyzing process is performed.

Next, after the point sequence information, the coefficients of thefirst fundamental form, and the coefficients of the second fundamentalform, which were generated and extracted at step S1 and step S2, havebeen collected (Yes in step S7), the CPU performs the curved surfacedata transferring process (step S9). On the other hand, in the casewhere such information has not been completed, the database evaluatingprocess is performed (No in step S7). That is, the shape reproducedbased on the principal direction, the reference position (point, line,or the like), the transformed amount, that were calculated at steps S4to S6, and the shape reproduced based on the point sequence information,the coefficients of the first fundamental form, and the coefficients ofthe second fundamental form are compared, and in the case where theymatch (Yes in step S8), the curved surface data transferring process isperformed (step S9). In the case where they do not match (No in stepS8), an accuracy improving process by approximation and interpolation isperformed. That is, the initial curved surface is approximated andinterpolated so that it becomes doubly differentiable, and the processesdescribed above are performed again from step S1. Then, at the stagewhere the comparative evaluation in step S8 is matched, the flow shiftsto the curved surface data transferring process.

The curved surface data is transferred to the converting program 2 orthe reproducing program 3 shown in FIG. 1. If the CPU receives a convertcommand, it executes the converting program 2. That is, firstly assumingthat a point selected as a feature point or a feature line, where theGaussian curvature becomes zero, is the transformation reference, theline of curvature is expansion and contraction transformed by the girthlength in the line of curvature direction so that the mesh or the curvedsurface is reproduced. Then, a plurality of point sequences on thecurved surface are extracted from the reproduced mesh or curved surface,and the point sequences are converted according to a graphicalrepresentation algorithm in another computer aided design system. Theconverted graphics data is reproduced by the other computer aided designapplication 22 and then output to the graphic display processing section11. The graphic display processing section 11 performs graphic displayprocessing of the data output from the computer aided design application22 and outputs this to the display section 12. The display section 12receives the input of the display data and displays this.

Moreover, if the CPU receives a reproduction command, it executes thereproducing program 3. The reproducing program makes the CPU execute theprocesses in the converting program except for the converting process.That is, assuming that a point where the Gaussian curvature becomes zerois the transformation reference, the line of curvature is expansion andcontraction transformed by the girth length in the line of curvaturedirection so that the mesh or the curved surface is reproduced. Then,the reproduced graphics data is output to the graphic display processingsection 11, and after display processing, it is displayed in the displaysection 12.

As described above, according to the computer aided design system of thepresent embodiment, an effect can be obtained where a free-form surfacecan be analyzed, converted and reproduced while retaining C2 continuity.Therefore, an effect can be obtained where the utility of a computeraided design model can be greatly increased, and the efficiency of thedesign and production processes can be improved.

In the computer aided design system of the present embodiment, thedescription is for a series of processes related to free-form surfaceanalysis, conversion, and reproduction in a computer aided design model.However, the computer aided design system of the present invention isnot limited to this, and is applicable to a computer graphics system, ora system and a program which performs graphical representation using acomputer.

Moreover, in the computer aided design system of the present embodiment,as shown in FIG. 2, as a suitable example, a curved surface is dividedinto mesh points, and then standardized by the fundamental vectors Suand Sv, so that the free-form surface analysis, conversion, andreproduction are performed by a u, v parameter form which uses pointsequence information (u, v). However, the computer aided design systemof the present invention is not limited to this, and coordinate valuesfor (x, y, z) coordinate parameters may be used.

The computer aided design system described above contains a computersystem inside. Moreover steps of a series of processes related to theaforementioned free-form surface analysis, conversion, and reproductionare stored in a computer readable recording media in program format. Acomputer reads out and executes this program, to thereby perform theabove processes. Here, the computer readable recording media is forexample a magnetic disk, magneto-optical disk, CD-ROM, DVD-ROM,semiconductor memory, or the like. Moreover, the arrangement may be suchthat this computer program is delivered to a computer by a communicationline, and the computer which receives this delivery, executes theprogram.

1. A computer aided design system comprising: a point sequenceinformation extraction device which extracts a plurality of pointsequences on a curved surface; a dividing device which generates acurved surface from the point sequences using another computer aideddesign system, and divides the curved surface into a mesh having apredetermined number of mesh points; a first fundamental form computingdevice for computing coefficients of a first fundamental form at a meshpoint of the mesh, the coefficients of the first fundamental form beingdefined at the mesh point by first-order differential values of the meshpoint; a second fundamental form computing device for computingcoefficients of a second fundamental form at the mesh point, thecoefficients of the second fundamental form being defined at the meshpoint by a product of second-order differential values of the mesh pointand a normal vector of the mesh at the mesh point; a memory device whichstores the point sequence information, the coefficients of the firstfundamental form and the coefficients of the second fundamental form; aprincipal curvature computing device which computes a principalcurvature of the mesh point based on the coefficients of the firstfundamental form and the coefficients of the second fundamental form; aline of curvature computing device which computes a line of curvatureshowing a principal direction of the mesh based on the principalcurvature; a feature point/feature line analyzing device which extractsa point or a line which becomes a reference point or a reference line,respectively, of a transformation defined by changing patterns of one ormore feature quantities among five feature quantities showing featuresof the curved surface, the five feature quantities comprising a Gaussiancurvature and a mean curvature computed based on the principalcurvature, the principal direction, the line of curvature, and thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form; and a girth length computing device whichcomputes a girth length based on a curvature computed from thecoefficients of the first fundamental form and the coefficients of thesecond fundamental form.
 2. The computer aided design system accordingto claim 1, further comprising: a reproducing device which transformsthe line of curvature for the girth length in the line of curvaturedirection, with the point or the line as the transformation referencepoint or reference line, respectively, and reproduces a shape of themesh or the curved surface.
 3. The computer aided design systemaccording to claim 2, further comprising: a converting device whichextracts a plurality of point sequences on a curved surface from thereproduced shape of the mesh or the curved surface, and converts thepoint sequences according to a graphical representation algorithm inanother computer aided design system.
 4. The computer aided designsystem according to claim 1, wherein the principal curvature computingdevice computes the principal curvature of the mesh point by solving aquadratic equation that includes the coefficients of the firstfundamental form and the coefficients of the second fundamental form. 5.A computer aided design program stored in a non-transitorycomputer-readable recording medium for causing a computer to execute: apoint sequence information extraction process for extracting a pluralityof point sequences on a curved surface; a dividing process forgenerating a curved surface from the point sequences using anothercomputer aided design program, and dividing the curved surface into amesh having a predetermined number of mesh points; a first fundamentalform computing process for computing coefficients of a first fundamentalform at a mesh point of the mesh, the coefficients of the firstfundamental form being defined at the mesh point by first-orderdifferential values of the mesh point; a second fundamental formcomputing process for computing coefficients of a second fundamentalform at the mesh point, the coefficients of the second fundamental formbeing defined at the mesh point by a product of second-orderdifferential values of the mesh point and a normal vector of the mesh atthe mesh point; a storage process for storing the point sequenceinformation, the coefficients of the first fundamental form and thecoefficients of the second fundamental form; a principal curvaturecomputing process for computing a principal curvature of the mesh basedon the coefficients of the first fundamental form and the coefficientsof the second fundamental form; a line of curvature computing processfor computing a line of curvature showing a principal direction of themesh based on the principal curvature; a feature point/feature lineanalyzing process for extracting a point or a line which becomes areference point or a reference line, respectively, of a transformationdefined by changing patterns of one or more feature quantities amongfive feature quantities showing features of the curved surface, the fivefeature quantities comprising a Gaussian curvature and a mean curvaturecomputed based on the principal curvature, the principal direction, theline of curvature, and the coefficients of the first fundamental formand coefficients of the second fundamental form; and a girth lengthcomputing process for computing a girth length based on a curvaturecomputed from the coefficients of the first fundamental form and thecoefficients of the second fundamental form.
 6. The computer aideddesign program according to claim 5, for further causing a computer toexecute a reproducing process for transforming the line of curvature forthe girth length in the line of curvature direction, with the point orthe line as the transformation reference point or reference line,respectively, and reproducing a shape of the mesh or the curved surface.7. The computer aided design program according to claim 6, for furthercausing a computer to execute a converting process for extracting aplurality of point sequences on a curved surface from the reproducedshape of the mesh or the curved surface, and converting the pointsequences according to a graphical representation algorithm in anothercomputer aided design system.
 8. A computer aided design systemcomprising: a point sequence information extraction device whichextracts a plurality of point sequences on a curved surface; a dividingdevice which generates a curved surface from the point sequences usinganother computer aided design system, and divides the curved surfaceinto a mesh having a predetermined number of mesh points; a firstfundamental form computing device for computing coefficients of a firstfundamental form at a mesh point of the mesh, the coefficients of thefirst fundamental form being defined at the mesh point by first-orderdifferential values of the mesh point; a second fundamental formcomputing device for computing coefficients of a second fundamental format the mesh point, the coefficients of the second fundamental form beingdefined at the mesh point by a product of second-order differentialvalues of the mesh point and a normal vector of the mesh at the meshpoint; and a memory device which stores the point sequence information,the coefficients of the first fundamental form and the coefficients ofthe second fundamental form, wherein, in a case where a mesh point ofthe mesh is represented by S(u, v), the coefficients of the firstfundamental form at the mesh point represented by S(u, v) are E, F andG, such that the coefficients E, F and G are represented by thefollowings equations:E=Su²;F=Su×Sv; andG=Sv², and wherein Su=∂s/∂u and Sv=∂s/∂v.
 9. The computer aided designsystem according to claim 8, wherein the coefficients of the secondfundamental form at a mesh point of the mesh are represented by L, M andN, such that L, M and N are represented by the following equations:L=n×Suu;M=n×Suv; andN=n×Svv, wherein n denotes a normal vector of the mesh at the mesh pointwhere the coefficients of the second fundamental form are represented byL, M and N, and wherein Suu=(∂²s/∂u²), Suv=(∂s/∂v)×(∂s/∂u), andSvv=(∂²s/∂v²).
 10. A computer aided design program stored in acomputer-readable recording medium for causing a computer to execute: apoint sequence information extraction process for extracting a pluralityof point sequences on a curved surface; a dividing process forgenerating a curved surface from the point sequences using anothercomputer aided design program, and dividing the curved surface into amesh having a predetermined number of mesh points; a first fundamentalform computing process for computing coefficients of a first fundamentalform at a mesh point of the mesh, the coefficients of the firstfundamental form being defined at the mesh point by first-orderdifferential values of the mesh point; a second fundamental formcomputing process for computing coefficients of a second fundamentalform at the mesh point, the coefficients of the second fundamental formbeing defined at the mesh point by a product of second-orderdifferential values of the mesh point and a normal vector of the mesh atthe mesh point; and a storage process for storing the point sequenceinformation, the coefficients of the first fundamental form and thecoefficients of the second fundamental form, wherein, in a case where amesh point of the mesh is represented by S(u, v), the coefficients ofthe first fundamental form at the mesh point represented by S(u, v) areE, F and G, such that the coefficients E, F and G are represented by thefollowings equations:E=Su²;F=Su×Sv; andG=Sv², and wherein Su=∂s/∂u and Sv=∂s/∂v.
 11. The computer aided designprogram according to claim 10, wherein the coefficients of the secondfundamental form at a mesh point of the mesh are represented by L, M andN, such that L, M and N are represented by the following equations:L=n×Suu;M=n×Suv; andN=n×Svv, wherein n denotes a normal vector of the mesh at the mesh pointwhere the coefficients of the second fundamental form are represented byL, M and N, and wherein Suu=(∂²s/∂u²), Suv=(∂s/∂v)×(∂s/∂u), andSvv=(∂²s/∂v²).
 12. A computer graphics system comprising: a pointsequence information extraction device which extracts a plurality ofpoint sequences on a curved surface; a dividing device which generates acurved surface from the point sequences using another computer graphicssystem, and divides the curved surface into a mesh having apredetermined number of mesh points; a first fundamental form computingdevice for computing coefficients of a first fundamental form at a meshpoint of the mesh, the coefficients of the first fundamental form beingdefined at the mesh point by first-order differential values of the meshpoint; a second fundamental form computing device for computingcoefficients of a second fundamental form at the mesh point, thecoefficients of the second fundamental form being defined at the meshpoint by a product of second-order differential values of the mesh pointand a normal vector of the mesh at the mesh point; and a memory devicewhich stores the point sequence information, the coefficients of thefirst fundamental form and the coefficients of the second fundamentalform, wherein, in a case where a mesh point of the mesh is representedby S(u, v), the coefficients of the first fundamental form at the meshpoint represented by S(u, v) are E, F and G, such that the coefficientsE, F and G are represented by the followings equations:E=Su²;F=Su×Sv; andG=Sv², and wherein Su=∂s/∂u and Sv=∂s/∂v.
 13. The computer graphicssystem according to claim 12, wherein the coefficients of the secondfundamental form at a mesh point of the mesh are represented by L, M andN, such that L, M and N are represented by the following equations:L=n×Suu;M=n×Suv; andN=n×Svv, wherein n denotes a normal vector of the mesh at the mesh pointwhere the coefficients of the second fundamental form are represented byL, M and N, and wherein Suu=(∂²s/∂u²), Suv=(∂s/∂v)×(∂s/∂u), andSvv=(∂²s/∂v²).
 14. A computer graphics program stored on anon-transitory computer-readable recording medium for causing a computerto execute: a point sequence information extraction process forextracting a plurality of point sequences on a curved surface; adividing process for generating a curved surface from the pointsequences using another computer graphics program, and dividing thecurved surface into a mesh having a predetermined number of mesh points;a first fundamental form computing process for computing coefficients ofa first fundamental form at a mesh point of the mesh, the coefficientsof the first fundamental form being defined at the mesh point byfirst-order differential values of the mesh point; a second fundamentalform computing process for computing coefficients of a secondfundamental form at the mesh point, the coefficients of the secondfundamental form being defined at the mesh point by a product ofsecond-order differential values of the mesh point and a normal vectorof the mesh at the mesh point; and a storage process for storing thepoint sequence information, the coefficients of the first fundamentalform and the coefficients of the second fundamental form, wherein, in acase where a mesh point of the mesh is represented by S(u, v), thecoefficients of the first fundamental form at the mesh point representedby S(u, v) are E, F and G, such that the coefficients E, F and G arerepresented by the followings equations:E=Su²;F=Su×Sv; andG=Sv², and wherein Su=∂s/∂u and Sv=∂s/∂v.
 15. The computer graphicsprogram according to claim 14, wherein the coefficients of the secondfundamental form at a mesh point of the mesh are represented by L, M andN, such that L, M and N are represented by the following equations:L=n×Suu;M=n×Suv; andN=n×Svv, wherein n denotes a normal vector of the mesh at the mesh pointwhere the coefficients of the second fundamental form are represented byL, M and N, and wherein Suu=(∂²s/∂u²), Suv=(∂s/∂v)×(∂s/∂u), andSvv=(∂²s/∂v²).